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Solve the system with three variables by Cramer's Rule.
-2x -y -3z = 3
2x -3y +z = -13
2x -3z = -11
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Your matrix
X1 X2 X3 b
1 -2 -1 -3 3
2 2 -3 1 -13
3 2 0 -3 -11
Write down the main matrix and find its determinant
X1 X2 X3
1 -2 -1 -3
2 2 -3 1
3 2 0 -3
Δ = -44
Replace the 1st column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 3 -1 -3
2 -13 -3 1
3 -11 0 -3
Δ1 = 176
Replace the 2nd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 -2 3 -3
2 2 -13 1
3 2 -11 -3
Δ2 = -88
Replace the 3rd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 -2 -1 3
2 2 -3 -13
3 2 0 -11
Δ3 = -44
x1 = Δ1 / Δ = 176 / (-44) = -4
x2 = Δ2 / Δ = (-88) / (-44) = 2
x3 = Δ3 / Δ = (-44) / (-44) = 1
Solution set:
x1 = -4
x2 = 2
x3 = 1
